Your problem is here:
'exclude' => '$user_id,1',
This will generate the string ‘$user_id,1’, it won’t insert the user ID into your string.
The reason for this is the difference between ‘$hello’ and “$hello” E.g.
$hello = 5;
echo '$hello'; // prints out $hello
echo "$hello"; // prints out 5
So using double rather than single quotes would fix your issue.
An even more reliable way
Use string concatenation to tie them together instead, e.g:
'exclude' => $user_id.',1',
The . operator joins two strings together, e.g.:
echo 'Hello there '.$firstname.', my name is '.$myname;
A more generic way
Use an array, and use implode to generate the string
$excluded_users = array();
$excluded_users[] = 1;
$excluded_users[] = $user_id;
// .. etc
'exclude' => implode( ',' , $excluded_users )
I recommend you read up on magic quotes in basic PHP and string handling